We begin by considering a non-adversarial detection formulation based on quantum hypothesis testing introduced in [1]}. Suppose that there is an unknown target quantum system characterized by a density operator \(\rho \in \mathcal {S}(\mathcal {H})\) . We consider the binary hypothesis testing scenario, assuming that the nature of the system specifies two possible choices for \(\rho =\rho _0\)  or \(\rho =\rho _1\) , each of which forms a hypothesis \(H_0,H_1\)  as follows:
\(H_0: \rho _0= \sum _{j = 1}^d{r^0_j|\psi _{j}\rangle \langle \psi _{j}|},\;\;H_1: \rho _1 = \sum _{i = 1}^d{r^1_i|\varphi _i\rangle \langle \varphi _i|}, \) 
